# 2019年IAO理论高年组第4题-彗星粒子

## 英文原题

4. Comet particles. Particles of characteristic cometary matter of various sizes come off a comet. Estimate, the characteristic sizes D of the particles which are not ejected outside the Solar System due to the solar radiation pressure.

Note: You will get more points for the solution if you first derive the algebraic formula of the answer D=f(a,b,c,d,e...) and only then get the numerical answer by inserting the numerical data a, b, c, d, e... into this formula.

## 中文翻译

4. 彗星粒子。一些典型的彗星物质构成的粒子从彗星上脱落下来。估计，不被太阳的辐射压弹射出太阳系的粒子的典型直径。

## 解答

（注意：本题目与标答不完全一致）

$$F_p=2\pi r^2 \frac{L}{4\pi R^2} \frac{\frac{h\nu}{c}}{h\nu}=\frac{L r^2}{2c R^2}$$

$$F=F_g -F_p =\frac{GMm}{R^2}-\frac{L r^2}{2c R^2}=\frac{GMm}{R^2}-\frac{Lr^2}{2cGMm}\frac{GMm}{R^2}=(1-\frac{Lr^2}{2cGMm})F_g$$

$$E_p +E_k=-(1-\frac{Lr^2}{2cGMm})\frac{GMm}{R}+\frac{1}{2}mv^2=0$$

$$E_p +E_k=-(1-\frac{Lr^2}{2cGMm})\frac{GMm}{R}+GMm(\frac{1}{R}-\frac{1}{2a})=0$$

$$(\frac{Lr^2}{2cGMm}-1)\frac{GMm}{R}+\frac{GMm}{R}-\frac{GMm}{2a}=0$$

$$\frac{Lr^2}{2cGMm}\frac{1}{R}-\frac{1}{2a}=0$$

$$\frac{Lr^2}{2cGM(\frac{4}{3}\pi\rho r^3 )}\frac{1}{R}-\frac{1}{2a}=0$$

$$\frac{L}{2cGM(\frac{4}{3}\pi\rho r)}\frac{1}{R}-\frac{1}{2a}=0$$